Assignment 2: Equations of Motion
A space shuttle sits on the launch pad for 2.0 minutes, and then goes from rest to 4600 m/s in 8.0 minutes. Treat its motion as straight-line motion. What is the average acceleration of the shuttle (a) during the first 2.0 minutes, (b) during the 8.0 minutes the shuttle moves, and (c) during the entire 10 minute period?
(a) 0 m/s2 (b) 9.6 m/s2 (c) 7.7 m/s2
In 1271, Marco Polo departed Venice and traveled to Kublai Khan's court near Beijing, approximately 7900 km away in a direction we will call positive. Assume that the Earth is flat (as some did at the time) and that the trip took him 4.0 years, with 365 days in a year. (a) What was his average velocity for the trip, in meters per second? (b) A 767 could make the same trip in about 9.0 hours. What is the average velocity of the plane in meters per second?
(a) 0.063 m/s (b) 240 m/s
The tortoise and the hare start a race from the same starting line, at the same time. The tortoise moves at a constant 0.200 m/s, and the hare at 5.00 m/s. (a) How far ahead is the hare after five minutes? (b) How long can the hare then snooze until the tortoise catches up
(a) 1440 m (b) 7200 s
A strange number line is measured in meters to the left of the origin, and in kilometers to the right of the origin. An object moves from −2200 m to 3.1 km. Find its displacement (a) in kilometers, and (b) in meters.
(a) 5.3 km (b) 5300 m
A diver is standing on a diving board that is 15 meters above the surface of the water. He jumps 0.50 meters straight up into the air, dives into the water and goes 3.0 meters underwater before returning to the surface. (a) Assume that "up" is the positive direction. What is the vertical displacement of the diver from when he is standing on the diving board to when he emerges from underwater? (b) Now assume that "down" is the positive direction. Determine the displacement, as before.
(a) −15 m (b) 15 m
To estimate the distance you are from a lightning strike, you can count the number of seconds between seeing the flash and hearing the associated thunderclap. For this purpose, you can consider the speed of light to be infinite (it arrives instantly). Sound travels at about 343 m/s in air at typical surface conditions. How many kilometers away is a lightning strike for every second you count between the flash and the thunder?
0.343 km
A photographer wants to take a picture of a particularly interesting flower, but he is not sure how far away to place his camera. He takes three steps forward, four back, seven forward, then five back. Finally, he takes the photo. As measured in steps, what was his displacement? Assume the forward direction is positive.
1
Anita and Nick are playing tug-of-war near a mud puddle. They are each holding on to an end of a taut rope that has a knot exactly in the middle. Anita's position is 6.2 meters east of the center of the puddle and Nick's position is 3.0 meters west of the center of the puddle. What is the location of the knot relative to the center of the puddle? Treat east as positive and west as negative.
1.6 meters
A rail gun uses electromagnetic energy to accelerate objects quickly over a short distance. In an experiment, a 2.00 kg projectile remains on the rails of the gun for only 2.10e−2 s, but in that time it goes from rest to a velocity of 4.00×103 m/s. What is the average acceleration of the projectile?
1.90e+5 m/s^2
A clown is shot straight up out of a cannon. The graph of his velocity versus time is shown. Determine the clown's vertical displacement from the instant he is shot out of the cannon at 0.0 seconds until when he reaches zero velocity at 2.5 seconds.
31 m
This problem requires you to apply some trigonometry. A friend of yours is 51.0 m directly to your left. You and she start running at the same time, and both run in straight lines at constant speeds. You run directly forward at 5.00 m/s for 125 m. She runs to the same final point as you, and wants to arrive at the same moment you do. How fast must she run?
5.40 m/s
The speed limit on a particular freeway is 28.0 m/s (about 101 km/hour). A car that is merging onto the freeway is capable of accelerating at 2.25 m/s2. If the car is currently traveling forward at 13.0 m/s, what is the shortest amount of time it could take the vehicle to reach the speed limit?
6.67 s
A vehicle is speeding at 115 km/h on a straight highway when a police car moving at 145 km/h enters the highway from an onramp and starts chasing it. The speeder is 175 m ahead of the police car when the chase starts, and both cars maintain their speeds. How much time, in seconds, elapses until the police car overtakes the speeder?
Distance = Velocity * Time Let D = distance travelled by speeding car before the police overtakes it T = time when the police car overtakes the first car For the first car, D = 115T --- call this Equation 1 and for the second (police) car, 0.175 + D = 145T --- call this Equation 2 Since D = 115T (from Equation 1), then Equation 2 simplifies to 0.175 + 115T = 145T Solving for "T" T = (0.175/30) * 3600 T = 21 seconds Hope this helps.
An airplane starts out at the Tokyo Narita Airport (x = 0) destined for Bangkok, Thailand (x = 4603 km). After departing from the gate, the plane has to wait ten minutes on the runway before taking off. When the plane is halfway to Bangkok, and cruising at its top speed, which of these is greater: instantaneous velocity, or average velocity since leaving the gate?
Instantaneous velocity
Suppose you drop a ball onto the ground (starting it with zero initial velocity). The ball bounces back upward with half the speed at which it hit the ground. How high up does the ball bounce back, as a percentage of the initial height?
Other answers missed "half the speed" start from height h, then v = √(2gh) v₁ = √(2gh₁) going up, speed is halved v₂ = (v₁/2) = √(2gh₂) combining (1/2)√(2gh₁) = √(2gh₂) (1/4)(2gh₁) = (2gh₂) (1/4)(h₁) = (h₂) h₂ = (1/4)(h₁) so it bounces back to 1/4 the height 25 percent!!!
To get a check to bounce 0.010 cm in a vacuum, it must reach the ground moving at a speed of 6.7 m/s. At what velocity toward the ground must you throw it from a height of 1.4 meters in order for it to it have a speed of 6.7 m/s when it reaches the ground? Treat downward as the negative direction, and watch the sign of your answer.
Vf^2 = Vi^2 + 2 . g . x 44.89 = Vi^2 + 2 *-9.8 * -1.4 Vi^2 = 44.89 - 27.44 = 17.45 Vi = - 4.2 m/s
The United States and South Korean soccer teams are playing in the first round of the World Cup. An American kicks the ball, giving it an initial velocity of 3.6 m/s. The ball rolls a distance of 5.0 m and is then intercepted by a South Korean player. If the ball accelerates at −0.50 m/s2 while rolling along the grass, find its velocity at the time of interception.
Vf^2 = Vi^2 + 2a*x 2.8 m/s
The city is trying to figure out how long the traffic light stay yellow at an intersection. The speed limit on the road is 45.0 km/h and the intersection is 23.0 m wide. A car is traveling at the speed limit in the positive direction and can brake with an acceleration of -5.20 m/s2. (a) If the car is to stop on the white line, before entering the intersection, what is the minimum distance from the line at which the driver must apply the brakes? (b) How long should the traffic light stay yellow so that if the car is just closer than that minimum distance when the light turns yellow, it can safely cross the intersection without having to speed up?
Vf^2 = Vi^2 + 2a*x a) 15m x= 1/2(Vi+Vf)t b)t= 3.68s
The brochure advertising a sports car states that the car can be moving at 100.0 km/h, and stop in 37.19 meters. What is its average acceleration during a stop from that velocity? Express your answer in m/s2. Consider the car's initial velocity to be a positive quantity.
Vf^2 = Vi^2 + 2a*x a= 10.39
A croissant is dropped from the top of the Eiffel Tower. The height of the tower is 300.5 meters (ignoring the antenna, and this figure changes slightly with temperature). Ignoring air resistance, at what speed will the croissant be traveling when it hits the ground?
Vf^2 = Vi^2 + 2ax 76.7 m/s
A watermelon cannon fires a watermelon vertically up into the air at a velocity of +8.00 m/s, starting from an initial position 1.20 meters above the ground. When the watermelon reaches the peak of its flight, what is (a) its velocity, (b) its acceleration, (c) the elapsed time, and (d) its height above the ground?
When the watermelon reaches its peak it is no longer moving up and has not yet started moving down. For an instant it is stationary. So, (a) its velocity is 0 m/s and (b) its acceleration is 0 m/s�. To find the elapsed time divide the velocity by the acceleration of gravity. t = v/a = 8/9.8 = 0.81 s (or c), (only when assuming acceleration is constant) b) acceleration is always -9.8 on earth d = x + v*t + 1/2*a*t^2 = 1.2 + 8*0.81 -1/2*9.8*0.81^2 = calculate for (d) 4.5m
Can a car with negative velocity move faster than a car with positive velocity? Explain.?
Yes, it can. A negative velocity is just moving backward. It can move as fast as you can make it go either way. Now the speed is relative, so which is negative depends on where you are looking from.
Can an object be increasing in speed as its acceleration decreases? If so, give an example. If not, explain.
Yes. Remember that acceleration is a change in velocity per unit time, or a rate of change in velocity. So, velocity can be increasing while the rate of increase goes down. For example, suppose a car is traveling at 40 km/h and a second later is going 50 km/h. One second after that, the car's speed is 55 km/h. The car's speed was increasing the entire time, but its acceleration in the second time interval was lower than in the first time interval.
Engineers are designing a rescue vehicle to catch a runaway train. When the rescue vehicle is launched from a stationary position, the train will be at a distance d meters away, moving at constant velocity v meters per second. The rescue vehicle needs to reach the train in t seconds. Write an equation for the constant acceleration needed for the rescue vehicle in terms of d, v, and t.
a = 2(vt + d)/t^2
You are driving in one direction on a long straight road. You drive in the positive direction at 126 km/h for 30.0 minutes, at which time you see a police car with someone pulled over, presumably for speeding. You then drive in the same direction at 100 km/h for 45.0 minutes. (a) How far did you drive? (b) What was your average velocity in kilometers per hour?
a) (126 km/hr) x (0.5 hr) + (100 km/hr) x (0.75 hr) = 138 km total (b) (138 km) / (0.5 + 0.75) hr = 110.4 km/hr
The school bus picks up Brian in front of his house and takes him on a straight-line 2.1 km bus ride to school in the positive direction. He walks home after school. If the front of Brian's house is the origin, (a) what is the position of the school, (b) what is his displacement on the walk home, and (c) what is his displacement due to the combination of the bus journey and his walk home?
a. The position of the school is +2.1 km. b. His displacement on the walk home is position of home minus position of the school (df - di) = 0 - 2.1 km = -2.1 km c. He ends up where he started, so his displacement is 0.
On a planet that has no atmosphere, a rocket 14.2 m tall is resting on its launch pad. Freefall acceleration on the planet is 4.45 m/s2. A ball is dropped from the top of the rocket with zero initial velocity. (a) How long does it take to reach the launch pad? (b) What is the speed of the ball just before it reaches the ground?
a. d = 0.5gt^2, 14.2 = 0.5 * 4.45 * t^2, 14.2 = 2.225t^2, t^2 = 14.2 / 2.225 = 6.38, t = sqrt(6.38) = 2.53 s. b. V^2 = 2gd, V^2 = 2 * 4.45 * 14.2 = 126.4, V = sqrt(126.4) = 11.2 m/s.
velocity
expresses an object's speed and direction, as in "three meters per second west." Velocity has a direction. In one dimension, motion in one direction is represented by positive numbers, and motion in the other direction is negative.
You are a bungee jumping fanatic and want to be the first bungee jumper on Jupiter. The length of your bungee cord is 45.0 m. Freefall acceleration on Jupiter is 23.1 m/s2. What is the ratio of your speed on Jupiter to your speed on Earth when you have dropped 45.0 m? Ignore the effects of air resistance and assume that you start at rest.
g=d/t^2 9.8m/s^2=45m/t^2 t(earth)=45m^1/2/9.8m/s^2^1/2=2.14s t(jupiter)=45m^1/2/23.1m/s^2^1/2=1.39s v=d/t v(earth)=45m/2.14s=21m/s v(jupiter)=45m/1.39s=32.4m/s 32.4m/s/21ms=1.54
Acceleration
is a change in velocity. Like velocity, it has a direction and in one dimension, it can be positive or negative. Average acceleration is the change in velocity divided by the elapsed time, and instantaneous acceleration is the acceleration of an object at a specific moment.
Displacement
is a measure of the change in the position of an object. It includes both the distance between the object's starting and ending points, and the direction from the starting point to the ending point. An example of displacement would be "three meters west" or "negative two meters".
average velocity
is its displacement divided by the elapsed time. In contrast, its instantaneous velocity is its velocity at a particular moment. This equals the displacement divided by the elapsed time for a very small interval of time, as the time interval gets smaller and smaller.
Position
is the location of an object relative to a reference point called the origin, and is specified by the use of a coordinate system.
Acceleration is the change in velocity with respect to time. The change in acceleration with respect to time is sometimes called the "jerk." What are its units?
m/s3
If you know only the initial position, the final position, and the constant acceleration of an object, can you calculate the final velocity? Explain.
no
Free-fall acceleration,
represented by g, is the magnitude of the acceleration due to the force of Earth's gravity. Near the surface of the Earth, falling objects have a downward acceleration due to gravity of 9.80 m/s2.
You stand near the edge of Half Dome in Yosemite, reach your arm over the railing, and (thoughtlessly, since what goes up does come down and there are people below) throw a rock upward at 8.00 m/s. Half Dome is 1460 meters high. How long does it take for the rock to reach the ground? Ignore air resistance.
t1= Vi(t) + 1/2*a*(t)^2 = 1.63s -1460=8t+1/2(-9.81)t^2 -4.9t^2+8t+1460 t2= quadratic formula = 16.46s t1+t2= t t=18.097secs
Two spacecraft are 13,500 m apart and moving directly toward each other. The first spacecraft has velocity 525 m/s and accelerates at a constant −15.5 m/s2. They want to dock, which means they have to arrive at the same position at the same time with zero velocity. (a) What should the initial velocity of the second spacecraft be? (b) What should be its constant acceleration?
used vf = vi + at to get 33.87 s -for time for the 1st aircraft used displacement = 1/2(vi + vf)t to get 8891 m -then 13500m - 8891m = 4609 m for distance traveled for the 2nd aircraft used displacement = 1/2(vi + vf)t to get 272 m for the vi of the 2nd aircraft (answer to a) -then used vf = vi + at to get -8.04 m/s^2 for acceleration of the 2nd aircraft (answer to b)
A Honda® and a Porsche® race, starting from the same point. The Honda accelerates at a constant 4.00 m/s2; the Porsche at a constant 8.00 m/s2. The Porsche gives the Honda an advantage by letting it start first. The Honda accelerates, and when it is traveling at 23.0 m/s, the Porsche starts. How far do the cars travel from the starting point before the Porsche catches up with the Honda?
v(t) = a t (so t = v/a) From this we deduce that the head start lasted t = 23.0m/s / (4.00 m/s^2) =5.75 seconds The Porsche's equations are (capital S and V and A for the Porsche) for t>5.75s: S(t) = 1/2 A (t-5.75s)^2 [Note that we shift the time by the delay, because at t=5.75s the Porsche won't have moved] We still have for the Honda s(t) = 1/2 a t^2 At what time is S(t) = s(t)? 1/2 A (t-5.75s)^2 = 1/2 a t^2 ( t - 5.75 ) = sqrt(a/A) t t (1 - sqrt(a/A) ) = 5.75 t = 5.75/(1-sqrt(a/A)) = 19.63 s And the point of overtaking therefore is s = 1/2 a t^2 = 1/2 * 4.00 m/s^2 * ( 19.63 s)^2 = 771 m
To determine freefall acceleration on a moon with no atmosphere, you drop your handkerchief off the roof of a baseball stadium there. The roof is 113 meters tall. The handkerchief reaches the ground in 18.2 seconds. What is freefall acceleration on this moon? (State the result as a positive quantity.)
x = Vi(t) + 1/2 a(t)^2 a= 0.682
On the Apollo 15 space mission, Commander David R. Scott verified Galileo's assertion that objects of different masses accelerate at the same rate. He did so on the Moon, where the acceleration due to gravity is 1.62 m/s2 and there is no air resistance, by dropping a hammer and a feather at the same time. Assume they were 1.25 meters above the surface of the Moon when he released them. How long did they take to land?
x = Vi(t) + 1/2*a*(t)^2 1.24 s
David is driving a minivan to work, and he is stopped at a red light. The light turns green and David drives to the next red light, where he stops again. Is David's average acceleration from light to light positive, negative, or zero?
zero
A person throws a ball straight up. He releases the ball at a height of 1.75 m above the ground and with a velocity of 12.0 m/s. Ignore the effects of air resistance. (a) How long until the ball reaches its highest point? (b) How high above the ground does the ball go?
𝑣 = 𝑣0 − 𝑔𝑡 = 0 𝑡 =𝑣0/𝑔 = 12/9.8 = 1.22 s ≈ 1.22 s ℎ = ℎ0 + 𝑣0𝑡 −1/2𝑔𝑡^2 = 1.75 + 12 ∗ 1.22 − 9.8 *1/2 * 1.22^2 ≈ 9.1 𝑚.